- Open Access
Some properties of meromorphically multivalent functions
© Xu et al.; licensee Springer. 2012
- Received: 18 October 2011
- Accepted: 16 April 2012
- Published: 16 April 2012
By using the method of differential subordinations, we derive certain properties of meromorphically multivalent functions.
2010 Mathematics Subject Classification: 30C45; 30C55.
- analytic function
- meromorphically multivalent function
Let f(z) and g(z) be analytic in U. Then, we say that f(z) is subordinate to g(z) in U, written f(z) ≺ g(z), if there exists an analytic function w(z) in U, such that |w(z)| ≤ |z| and f(z) = g(w(z)) (z ∈ U). If g(z) is univalent in U, then the subordination f(z) ≺ g(z) is equivalent to f(0) = g(0) and f(U) ⊂ g(U).
Recently, several authors (see, e.g., [1–7]) considered some interesting properties of meromorphically multivalent functions. In the present article, we aim at proving some subordination properties for the class Σ(p).
To derive our results, we need the following lemmas.
Our first result is contained in the following.
The bound β is the best possible for each .
for all z ∈ U because of g(δ) = 0. This proves (2.3).
Hence, we conclude that the bound β is the best possible for each .
Next, we derive the following.
The bound in (2.14) is sharp.
from (2.16), we get the inequality (2.14).
as z → -1. Now the proof of the theorem is complete.
Finally, we discuss the following theorem.
But both (2.23) and (2.24) contradict the assumption (2.17). Therefore, we have Rep(z) > 0 for all z ∈ U. This shows that (2.18) holds true.
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